Member-only story
Stochastic Differential Equations —The Ornstein-Uhlenbeck Process
I haven’t written for a few months and while I was previously writing about optimization and linear programming, this article will not be about that. This article is about the basics of stochastic differential equations and something called the Ornstein-Uhlenbeck process. It is also called a “mean-reverting process” and it can be considered a modification of the “random-walk” where the particle tends to drift toward the mean of the process.
The article will be broken into a few parts which will go over some basics then the Ornstein-Uhlenbeck process specifically and I’ll write some code to simulate a realization of the process. I’ll assume that you have some background in differential equations and probability theory but it doesn’t need to be too extensive.
The Motivation
Generally when you take a course in differential equations you begin with separable equations and you may see something like exponential decay or growth which are relatively straight forward and easy to solve. An example would be something like logistic growth (also called Verhulst’s equation). Suppose we have some bacteria in a Petri dish and we know several things (I’ll know these are completely made up and kind of ridiculous but it’ll be obvious why I chose them shortly).
- The population starts at 100 bacteria.
- The population is 200 after an hour.
- The population has a carrying capacity (maximum…